In words, explain what is measured by each of the following:

In words, explain what is measured by each of the following:

A. The Sum of Squares (ss)

SS is a tool used in statistics. The SS is the sum of the squared deviations between a set of observations and a reference set, or it is an estimate of the variability within a set of samples. This statistical tool, particularly when used in regression analysis, displays how well the data match the model or target.

B. Variance

Variance, on the other hand, is a statistical measure used to represent the average deviation from the mean of a random variable. Variance is calculated by summing the squared difference between each observation and its average value (mean). The variance indicates how much variation or dispersion there is in a set of observations about their average. Lower values mean less variability and higher values indicate more variability.

C. Standard Deviation

Standard deviation is a statistical measure used to represent the average deviation from the mean of a set of values. It is often used as a measure of how far data points tend to be scattered about their average value. The standard deviation is the square root of the variance, and for that reason, it is also referred to as the standard error.

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2. Can SS ever have a value less than zero? Explain your answer. 

No, SS cannot be less than zero considering that it is calculated by adding squared variances. There is never a squared deviation that is 0 or less.

3. Is it possible to obtain a negative value for the variance or the standard deviation? 

Variance cannot be negative, the lowest value taken by variance is zero, if we add an observation that is equal to the mean, variance will be zero.

Standard deviation cannot be negative because the smallest standard deviation that can exist is zero. The least practicable value’s standard deviation should be zero. The standard deviation must be greater than 0 and positive if you are not roughly equal to at least two figures in your data collection.

4. What does it mean for a sample to have a standard deviation of zero? Describe the scores in such a sample. 

There is no variation when the standard deviation is zero. In this instance, the values of every score in the sample are identical to one another. If the scores are truly zero, it is not possible for there to be any variability in the sum of scores. 

5. Explain why the formulas for sample variance and population variance are different. 

The term “variance” refers to the mean deviation, which is calculated for a population as the total of deviations by N. Since the sample variance will be biased, the corresponding population value will always be understated because of sampling error. The difference between sample and population variance is also due to sampling error.

6. Why is variance calculated by squaring the deviations?

Squares are used to calculate variance because they give outliers a higher weight than data that is more representative of the population as a whole. This means that if we were to calculate the sum of squared deviations for every random variable, then squaring them would lead to a sum identical to the average and variance for the population of observations.

7. What is the Explained Sum of Squares?

Explained Sum of Squares is a measure of how accurately a model, frequently a regression model, reproduces the data it is intended to model. Particularly, the explained sum of squares measures the degree of variation in the modeled values and compares it to the total sum of squares (TSS), which assesses the degree of variation in the observed data, and to the residual sum of squares, which assesses the degree of variation in the error between the observed data and modeled values.

The formula for ESS is:

8. Types of Sum of Squares (SS)

There are three types of sum of squares in regression analysis. They include residual sum of squares, regression sum of squares, and total sum of squares.

1. Residual Sum of Squares

In essence, the residual sum of squares gauges the variability of modeling errors. In other words, it illustrates how the model cannot account for the variation in the dependent variable in a regression analysis. A low residual sum of squares shows that the model and the dependent variable are well modeled. Furthermore, a high residual sum of squares indicates that the data are not well modeled by the model.

The formula for this model is:

Where:

ŷ_i – the estimate of the regression line’s value

y_i – the observed value

2. Regression Sum of Squares

A regression model’s ability to accurately represent the modeled data is indicated by the regression sum of squares. High regression sum of squares shows the model cannot accurately represent the data. Furthermore, a low regression sum of squares shows that the model can accurately represent the data.

The formula for this model is:

Where:

ȳ – the mean value of a sample

ŷ_i – the value estimated by the regression line       

3. Total Sum of Squares

The total sum of squares measures all aspects of regression analysis, including errors in modeling and unexplained variability in dependent variable(s). It is a measure that indicates how well models explain variations in dependent variables and holds for other comparisons between models and models and observations. As such, it measures a model’s ability to account for variation within and between groups.

The formula for this model is:

Where:

ȳ – the mean value of a sample

y_i – the value in a sample